Werner state
A Werner state[1] is a × -dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form . That is, it is a bipartite quantum state that satisfies
for all unitary operators U acting on d-dimensional Hilbert space. These states were first developed by Reinhard F. Werner in 1989.
General definition
Every Werner state is a mixture of projectors onto the symmetric and antisymmetric subspaces, with the relative weight being the main parameter that defines the state, in addition to the dimension :
where
are the projectors and
is the permutation or flip operator that exchanges the two subsystems A and B.
Werner states are separable for p ≥ 1⁄2 and entangled for p < 1⁄2. All entangled Werner states violate the PPT separability criterion, but for d ≥ 3 no Werner state violates the weaker reduction criterion. Werner states can be parametrized in different ways. One way of writing them is
where the new parameter α varies between −1 and 1 and relates to p as
Two-qubit example
Two-qubit Werner states, corresponding to above, can be written explicitly in matrix form as
Equivalently, these can be written as a convex combination of the totally mixed state with (the projection onto) a Bell state:
where (or, confining oneself to positive values, ) is related to by . Then, two-qubit Werner states are separable for and entangled for .
Werner-Holevo channels
A Werner-Holevo quantum channel with parameters and integer is defined as [2] [3] [4]
where the quantum channels and are defined as
and denotes the partial transpose map on system A. Note that the Choi state of the Werner-Holevo channel is a Werner state:
where .
Multipartite Werner states
Werner states can be generalized to the multipartite case.[5] An N-party Werner state is a state that is invariant under for any unitary U on a single subsystem. The Werner state is no longer described by a single parameter, but by N! − 1 parameters, and is a linear combination of the N! different permutations on N systems.
References
- Reinhard F. Werner (1989). "Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model". Physical Review A. 40 (8): 4277–4281. Bibcode:1989PhRvA..40.4277W. doi:10.1103/PhysRevA.40.4277. PMID 9902666.
- Reinhard F. Werner and Alexander S. Holevo (2002). "Counterexample to an additivity conjecture for output purity of quantum channels". Journal of Mathematical Physics. 43 (9): 4353–4357. arXiv:quant-ph/0203003. Bibcode:2002JMP....43.4353W. doi:10.1063/1.1498491. S2CID 42832247.
- Fannes, Mark; Haegeman, B.; Mosonyi, Milan; Vanpeteghem, D. (2004). "Additivity of minimal entropy out- put for a class of covariant channels". unpublished. arXiv:quant-ph/0410195. Bibcode:2004quant.ph.10195F.
- Debbie Leung and William Matthews (2015). "On the power of PPT-preserving and non-signalling codes". IEEE Transactions on Information Theory. 61 (8): 4486–4499. arXiv:1406.7142. doi:10.1109/TIT.2015.2439953. S2CID 14083225.
- Eggeling, Tilo; Werner, Reinhard (2001). "Separability properties of tripartite states with UxUxU-symmetry". Physical Review A. 63: 042111. arXiv:quant-ph/0010096. doi:10.1103/PhysRevA.63.042111. S2CID 119350302.